= Solution
The <electromagnetic energy density> in <vacuum> is $u_E=\epsilon_0E^2/2$. For a length $\ell_z$ of the coaxial <capacitor>, integrate over annular shells:
$$
U=\int_{r_1}^{r_2}\frac{\epsilon_0}{2}\frac{V^2}{\eta^2r^2}(2\pi r\ell_z)\,dr
=\frac{\pi\epsilon_0\ell_zV^2}{\eta^2}\ln\frac{r_2}{r_1}.
$$
Since $C=C'\ell_z=2\pi\epsilon_0\ell_z/\eta$, this becomes
$$
\boxed{U=\frac12CV^2.}
$$
The integral is over the inter-cylinder <electric field>, and gives the stored <energy>, independently of which electrode is assigned zero <electric potential>.
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